and Theorem REF  implies the following characterization.
In \(X=\mathbb {R}^n\)  (equipped with the usual Euclidean distance and the Lebesgue measure),
the corresponding result in terms of Whitney cubes has been considered in [1]}
in the unweighted case \(\beta =0\) ,
but the weighted result is new even in \(\mathbb {R}^n\) .
Since \(N^{1,p}(\mathbb {R}^n)=W^{1,p}(\mathbb {R}^n)\)  and \(\mathbb {R}^n\)  supports \((1,s)\) -Poincaré inequalities for every \(s\ge 1\)  with \(\lambda =1\) ,
Theorem REF  follows from Corollary REF .
